Optimal. Leaf size=74 \[ \frac {2 b^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{5 a^{5/2}}+\frac {2 b^2 p}{5 a^2 x}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}-\frac {2 b p}{15 a x^3} \]
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Rubi [A] time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2455, 325, 205} \[ \frac {2 b^2 p}{5 a^2 x}+\frac {2 b^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{5 a^{5/2}}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}-\frac {2 b p}{15 a x^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 325
Rule 2455
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^6} \, dx &=-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}+\frac {1}{5} (2 b p) \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx\\ &=-\frac {2 b p}{15 a x^3}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}-\frac {\left (2 b^2 p\right ) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{5 a}\\ &=-\frac {2 b p}{15 a x^3}+\frac {2 b^2 p}{5 a^2 x}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}+\frac {\left (2 b^3 p\right ) \int \frac {1}{a+b x^2} \, dx}{5 a^2}\\ &=-\frac {2 b p}{15 a x^3}+\frac {2 b^2 p}{5 a^2 x}+\frac {2 b^{5/2} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{5 a^{5/2}}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 49, normalized size = 0.66 \[ -\frac {\log \left (c \left (a+b x^2\right )^p\right )}{5 x^5}-\frac {2 b p \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {b x^2}{a}\right )}{15 a x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 170, normalized size = 2.30 \[ \left [\frac {3 \, b^{2} p x^{5} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 6 \, b^{2} p x^{4} - 2 \, a b p x^{2} - 3 \, a^{2} p \log \left (b x^{2} + a\right ) - 3 \, a^{2} \log \relax (c)}{15 \, a^{2} x^{5}}, \frac {6 \, b^{2} p x^{5} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 6 \, b^{2} p x^{4} - 2 \, a b p x^{2} - 3 \, a^{2} p \log \left (b x^{2} + a\right ) - 3 \, a^{2} \log \relax (c)}{15 \, a^{2} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 71, normalized size = 0.96 \[ \frac {2 \, b^{3} p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{5 \, \sqrt {a b} a^{2}} - \frac {p \log \left (b x^{2} + a\right )}{5 \, x^{5}} + \frac {6 \, b^{2} p x^{4} - 2 \, a b p x^{2} - 3 \, a^{2} \log \relax (c)}{15 \, a^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.40, size = 235, normalized size = 3.18 \[ -\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{5 x^{5}}-\frac {-6 \sqrt {-a b}\, b^{2} p \,x^{5} \ln \left (-b x -\sqrt {-a b}\right )+6 \sqrt {-a b}\, b^{2} p \,x^{5} \ln \left (-b x +\sqrt {-a b}\right )-12 a \,b^{2} p \,x^{4}-3 i \pi \,a^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )+3 i \pi \,a^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}+3 i \pi \,a^{3} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}-3 i \pi \,a^{3} \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}+4 a^{2} b p \,x^{2}+6 a^{3} \ln \relax (c )}{30 a^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.53, size = 62, normalized size = 0.84 \[ \frac {2}{15} \, b p {\left (\frac {3 \, b^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, b x^{2} - a}{a^{2} x^{3}}\right )} - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 61, normalized size = 0.82 \[ \frac {2\,b^{5/2}\,p\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{5\,a^{5/2}}-\frac {\frac {2\,b\,p}{3\,a}-\frac {2\,b^2\,p\,x^2}{a^2}}{5\,x^3}-\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{5\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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